Air conditioning new building experiences?

Yes, these are two completely independent systems and both make sense in today's houses. If the money is available, then definitely have either one or the other or both installed.

Of course, it would be ideal if both work together and not against each other.

What do you mean by not playing against each other? What should I pay attention to here?

That the ventilation systems should run only basic ventilation during the heat periods and then switch back to normal or intensive preferably automatically when it gets cooler outside. Furthermore, attention must be paid to shading so that it also helps save energy. Presence also plays a role.

It is counterproductive to run the air conditioning at full load while pumping warm air from outside inside.

etc.

No, that is irrelevant. However, the system that should cool as much of the house as possible must be located at the top of the stairwell. In the bedrooms, it should preferably not blow directly onto the beds.